3.326 \(\int \frac{-1+x^4}{x^2 \sqrt{1+x^2+x^4}} \, dx\)

Optimal. Leaf size=16 \[ \frac{\sqrt{x^4+x^2+1}}{x} \]

[Out]

Sqrt[1 + x^2 + x^4]/x

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Rubi [A]  time = 0.0180566, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1590} \[ \frac{\sqrt{x^4+x^2+1}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^4)/(x^2*Sqrt[1 + x^2 + x^4]),x]

[Out]

Sqrt[1 + x^2 + x^4]/x

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

\begin{align*} \int \frac{-1+x^4}{x^2 \sqrt{1+x^2+x^4}} \, dx &=\frac{\sqrt{1+x^2+x^4}}{x}\\ \end{align*}

Mathematica [A]  time = 0.0360015, size = 16, normalized size = 1. \[ \frac{\sqrt{x^4+x^2+1}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^4)/(x^2*Sqrt[1 + x^2 + x^4]),x]

[Out]

Sqrt[1 + x^2 + x^4]/x

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Maple [A]  time = 0.006, size = 29, normalized size = 1.8 \begin{align*}{\frac{ \left ({x}^{2}+x+1 \right ) \left ({x}^{2}-x+1 \right ) }{x}{\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)/x^2/(x^4+x^2+1)^(1/2),x)

[Out]

(x^2+x+1)*(x^2-x+1)/(x^4+x^2+1)^(1/2)/x

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Maxima [A]  time = 1.15252, size = 30, normalized size = 1.88 \begin{align*} \frac{\sqrt{x^{2} + x + 1} \sqrt{x^{2} - x + 1}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/x^2/(x^4+x^2+1)^(1/2),x, algorithm="maxima")

[Out]

sqrt(x^2 + x + 1)*sqrt(x^2 - x + 1)/x

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Fricas [A]  time = 1.49778, size = 31, normalized size = 1.94 \begin{align*} \frac{\sqrt{x^{4} + x^{2} + 1}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/x^2/(x^4+x^2+1)^(1/2),x, algorithm="fricas")

[Out]

sqrt(x^4 + x^2 + 1)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)/x**2/(x**4+x**2+1)**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)*(x**2 + 1)/(x**2*sqrt((x**2 - x + 1)*(x**2 + x + 1))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} - 1}{\sqrt{x^{4} + x^{2} + 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/x^2/(x^4+x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((x^4 - 1)/(sqrt(x^4 + x^2 + 1)*x^2), x)